Order Form - FENOMEC

42 CHAPTER II. THE RIEMANN PROBLEM
Note that, by (4.1),
ϕ ♮ (u) ≤ ϕ ♯ (u) 0,
ϕ ♯ 0 (u) 0:
(a) If u r ≥ u l , the solution is a rarefaction connecting u l to u r .
(b) If u r ∈ [ )
ϕ ♯ (u l ),u l , the solution is a classical shock.
(c) If u r ∈ [ ϕ ♭ (u l ),ϕ ♯ (u l ) ) , the solution consists of a nonclassical shock connecting
u l to ϕ ♭ (u l ) followed by a classical shock.
(d) If u r ≤ ϕ ♭ (u l ), the solution consists of a nonclassical shock connecting u l to
ϕ ♭ (u l ) followed by a rarefaction connecting to u r .
In Cases (a), (b), and (d) the solution is monotone, while it is non-monotone in Case
(c). The classical Riemann solution (Theorem 2.2) is also admissible as it contains
only classical waves (for which (4.4) is irrelevant).
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Observe that the value ϕ ♯ (u l ) determines an important transition from a one-wave
to a two-wave pattern. The nonclassical Riemann solution fails to depend pointwise
continuously upon its initial data, in the following sense. The solution in Case (c)
contains the middle state ϕ ♭ (u l )whichdoes not converge to u l nor u r when the right